Graded identities and isomorphisms on algebras of upper block-triangular matrices

Let G be an abelian group and K an algebraically closed field of characteristic zero. A. Valenti and M. Zaicev described the G-gradings on upper block-triangular matrix algebras provided that G is finite. We prove that their result holds for any abelian group G: any grading is isomorphic to the tensor product A circle times B of an elementary grading A on an upper block-triangular matrix algebra and a division grading B on a matrix algebra. We then consider the question of whether graded identities A circle times B, where B is an algebra with a division grading, determine A circle times B up to graded isomorphism. In our main result, Theorem 3, we reduce this question to the case of elementary gradings on upper block-triangular matrix algebras which was previously studied by O.M. Di Vincenzo and E. Spinelli. (C) 2019 Elsevier Inc. All rights reserved.